Probability
Complements, impossible and sure events
Use P(not E) = 1 – P(E), and know why probabilities lie between 0 and 1.
'Not E' (written Ē) is everything that can happen except E. On a die, if E is 'a 6', then 'not E' is 1, 2, 3, 4 or 5. E and not E are called complementary events.
Between them they cover every outcome exactly once, so:
P(E) + P(not E) = 1, which means P(not E) = 1 – P(E)
Use it when 'not E' is easier to count, or as a check: work out a question both ways and see that the answers add to 1.
Example: Sangeeta wins a tennis match with probability 0.62. One of the two players must win, so P(Reshma wins) = 1 – 0.62 = 0.38.
| Event | Example | Probability |
|---|---|---|
| impossible | getting an 8 on a die | 0 |
| sure (certain) | a number less than 7 on a die | 1 |
| anything else | between 0 and 1 |
So a probability can never be negative or more than 1. Values like –1.5 or 1.2 are always wrong.
A pack of cards: 52 cards = 4 suits of 13 (hearts and diamonds are red, spades and clubs are black). Each suit has an ace, a king, a queen and a jack. The face cards are kings, queens and jacks: 12 in all. P(an ace) = 4/52 = 1/13, so P(not an ace) = 12/13.
'And' and 'or': 'a red king' needs both (2 cards). 'Red or a king' means either one, but don't count a card twice: 26 red + the 2 black kings = 28.
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